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33 Chapter 1 x Introduction Separating the variables, we may integrate: w ³ Zo dZ Z SPro4 5SP ro2 t º » ¬ 3mh sinT ¼ ª t 2hIo sinT ³0 dt, or: Z Z o exp « 1.54* A disk of radius R rotates at angular velocity : inside an oil container of viscosity P, as in Fig. P1.54. Assuming a linear velocity profile and neglecting shear on the outer disk edges, derive an expression for the viscous torque on the disk. Ans. Fig. P1.54 Solution: At any r d R, the viscous shear W | P:r/h on both sides of the disk. Thus, d(torque) dM 2rW dA w or: P1.55 M 4S P: h R ³ r 3 dr 0 2r P:r h 2Sr dr, SP:R4 h Ans A block of weight W is being pulled W over a table by another weight Wo, as shown in h Fig. P1.55. Find an algebraic formula for the steady velocity U of the block if it slides on an oil film of thickness h and viscosity P. The block bottom area A is in contact with the oil. Neglect the cord weight and the pulley friction. Wo Fig. P1.55 Solution: This problem is a lot easier to solve than to set up and sketch. For steady motion, there is no acceleration, and the falling weight balances the viscous resistance of the oil film: